Entanglement spectrum, the full spectrum of the reduced density matrix of a subsystem, plays a major role in characterising many-body quantum systems. In recent years, it has been widely studied in the fields of condensed matter physics, quantum information, high energy and black-hole physics. As first pointed out by Haldane and Li in the context of fractional quantum Hall effect, the entanglement spectrum can serve as fingerprint of topological order (TO), which is itself a non-local feature and a pattern of long-range entanglement. The correspondence between ES and TO has been further explored since then and the importance of ES has been extended to the context of quantum criticality, symmetry-breaking phases, tensor networks, eigenstate thermalization, and many-body localization.
While there has been a surge of theoretical works on the subject, no experimental measurement has been performed to this date, due to the lack of an implementable measurement scheme. In our recent preprint, we propose a measurement protocol to access the entanglement spectrum of many-body states in experiments with cold atoms in optical lattices. Our scheme effectively performs a Ramsey spectroscopy of the entanglement Hamiltonian, and is based on the ability to produce several copies of the state under investigation together with the possibility to perform a global swap gate between two copies conditioned on the state of an auxiliary qubit. We show how the required conditional swap gate can be implemented with cold atoms, either by using Rydberg interactions or coupling the atoms to a cavity mode. We illustrate these ideas on a simple (extended) Bose-Hubbard model where such a measurement protocol reveals topological features of the Haldane phase.